TAOCP 7.2.2.1 Exercise 338
The statement refers to six target shapes shown in Figure 338, but the figure itself is not included in the supplied material.
Section 7.2.2.1: Dancing Links
Exercise 338. [22] Show that there are exactly eight different tetracubes, polycubes of size 4. Which of the following shapes can they make, respecting gravity? How many solutions are possible?
Verified: no
Solve time: 2m23s
Solution
The statement refers to six target shapes shown in Figure 338, but the figure itself is not included in the supplied material. The names “twin towers”, “double claw”, “cannon”, “up 3”, “up 4”, and “up 5” do not uniquely determine the corresponding polycubes, because different coordinate sets can have the same informal descriptions.
The first part of the exercise can be completed independently. A tetracube is a connected union of four unit cubes. Up to rotations and reflections in space, there are exactly eight such polycubes. They are the straight tetracube, the square tetracube, the $T$ tetracube, the skew tetracube, the $L$ tetracube, the $J$ tetracube, the $S$ tetracube, and the three-dimensional corner tetracube. A complete enumeration follows by considering the possible shapes obtained after deleting one cube from a tetracube. The remaining three cubes must form one of the two tricubes, the straight tricube or the bent tricube. Attaching the fourth cube in every possible position and identifying rotations gives the eight classes above. No ninth class can occur, because every connected set of four cubes contains a spanning tree with three face adjacencies, and the two tricube cases exhaust the possible first three cubes.
The packing counts for the six named figures require the exact cell coordinates of the six figures in Figure 338. Those coordinates are missing here, so the requested statements “Which of the following shapes can they make” and “How many solutions are possible” cannot be derived uniquely from the supplied information.
Please provide the image of Figure 338 or the six target-shape coordinates, and the complete solution can then be given with the exact solution counts and a rigorous lower and upper verification.